The Qualitative Research For Two Classes Of Fractional Evolution Equations With Logarithmic Source Term

In this thesis,we mainly consider the initial-boundary value problem for two classes of fractional evolution equations with logarithmic source term.Where discussed:one is the fractional damped wave equation with logarithmic source term utt+(-Δ)su+(-Δ)sut=uln|u|,the other is the fractional pseudo-parabolic equation with memory term and logarithmic source term ut+(-Δ)su+(-Δ)sut=∫0t g(t-τ)(-Δ)su(τ)dτ+u ln |u|.We mainly study in detail the well-posedness of local solutions,the global existence,asymptotic behavior and blow up result.Chapter 1 introduces the research background,research status of fractional damped wave equations and fractional pseudo-parabolic equations,and the related preliminaries.Chapter 2 we focus on the initial-boundary value problem for a class of fractional damped wave equations with logarithmic source term.Firstly,the potential well theoretical framework is derived by introducing the total energy functional,potential energy functional,Nehari functional,and related lemmas.Then,using the Galerkin approximation and the contraction mapping principle,the local well-posedness of weak solution is obtained at arbitrary initial energy conditions.Furthermore,under the conditions of subcritical initial energy and critical initial energy:(1)we establish the global existence of the weak solution by apply the modified potential well theory and Galerkin approximation;(2)we combine perturbation energy method and integro-differential inequalities to prove the asymptotic behavior of solutions:polynomial and exponential energy decay estimates;(3)we utilize the concavity method,perturbation energy method and modified potential well to get the solutions blow up in infinite time,finite time and give the upper bounds for the blow up time.The main goal of the Chapter 3 is discussing the initial-boundary value problem for a class of fractional pseudo-parabolic equations with memory term and logarithmic source term.The modified potential well theory,the contraction mapping principle,Galerkin approximation,perturbation energy method,and concavity method are applied to prove that:(1)the local well-posedness of weak solution under arbitrary initial energy conditions;(2)the global existence,asymptotic behavior and the infinite time blow up of solutions under subcritical and critical initial energy conditions;(3)the global existence,asymptotic behavior and the finite time blow up phenomenon at high initial energy.